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Calculus I









© The scientific sentence. 2010

Calculus I:
Theorems of analysis for the continuous functions




1. Extrema of a function



2. Continuity of a function:


A function f(x) is continuous at x = a if

lim f(x) = f(a)
x → a


A function f(x) is continuous on the interval [a, b] if it is continuous at each point in the interval.

A closed interval includes its endpoints: [a,b] An open interval does not include its endpoints [a,b[ or [a,b)

Bounded intervals are finite intervals, that is of finite size. Conversely, a set which is not bounded is called unbounded. The set R of all reals is the only interval that is unbounded at both ends.



3. Derivative of f(x)

The derivative of f(x) with respect to x is the function f'(x), defined as,

f'(x) = lim (f(x + h) - f(x) )/ h
h → 0


It is also

f'(x) = lim (f(x) - f(a)) / (x - a)
x → a


df(x)/dx = dy/dx is the Leibniz's notation, f'(x) Lagrange notation,
Newton notation, and Dx is the Euler notation.



4. Function differentiable

A function f(x) is called differentiable at x = a if f'(a) exists, and f(x)is called differentiable on an interval if the derivative exists for each point in that interval.

Theorem

If f(x) is differentiable at x = a then f(x) is continuous at x = a.



5. Extreme value theorem:



if a fuction f(x) is continuous on a closed interval [a,b], then f(x) has both absolute maximun and minimum value on [a,b].

f cont [a,b] it exists c, d in [a,b]: f(c) ≤ f(x) ≤ f(d) for all x in [a,b]

In other words,

if a fuction f(x) is continuous on a closed interval [a,b], then there are two numbers a < c, d < b so that f(c) is an absolute maximum for the function and f(d) is an absolute minimum for the function.

Definition

f(x) has :

1. an absolute maximum at x = c if f(x) ≤ f(c) for every x in the domain considered.

2. a relative maximum at x = c if f(x) ≤ f(c) for every x in some open interval around x = c .

3. an absolute minimum at x = c if f(x) ≥ f(c) for every x in the domain considered.

4. a relative minimum at x = c if f(x) ≥ f(c)for every x in some open interval around x = c.

We say absolute or global, relative or local.



6. Intermediate Value Theorem



if f(x) is continuous on [a, b] and M is any number between f(a) and f(b), then there exists a number c such that:
f(c) = M , and a < c < b

This theorem is used to justify that a function has a root in an inerval.



7. Mean Value Theorem

mvt.png

if a function f(x) is continuous on the closed interval [a,b], differentiable on the open interval (a,b),
then there is a number c such that a < c < b and

f'(c) = (f(b) - f(a))/(b - a)



8. Rolle’s Theorem

if a function f(x) is continuous on the closed interval [a,b, differentiable on the open interval (a,b), f(a) = f(b)
then there is a number c such that a < c < b and

f'(c) = 0 (or f(x) has a critical point in (a,b)).



9. Fermat’s Theorem

if f(x) has a relative extrema at x = c and f'(c) exists then x = c is a critical point of f(x), such that f'(c) = 0.








  


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